Question

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.
lim _{x
ightarrow 9} \frac{x^{2}-7 x-18}{x-9}

Explanation:

Step1: Factor the numerator

The numerator \(x^{2}-7x - 18\) can be factored as \((x - 9)(x+2)\). So the limit becomes \(\lim_{x
ightarrow9}\frac{(x - 9)(x + 2)}{x - 9}\).

Step2: Simplify the function

Cancel out the common factor \((x - 9)\) (for \(x
eq9\)). The simplified function is \(y=x + 2\).

Step3: Evaluate the limit

Substitute \(x = 9\) into \(y=x + 2\). We get \(y=9+2\).

Answer:

\(11\)